Question 6
Write $\mathbb R^+$ for the positive real numbers. Let $\mathcal F$ be the set of all functions $f:\mathbb R^+\to\mathbb R^+$ satisfying
$$f(3x)\ge f(f(2x))+x$$
for every $x\gt 0$. Find the largest real constant $\alpha$ such that $f(x)\ge\alpha x$ for every $f\in\mathcal F$ and every $x\in\mathbb R^+$.
$$f(3x)\ge f(f(2x))+x$$
for every $x\gt 0$. Find the largest real constant $\alpha$ such that $f(x)\ge\alpha x$ for every $f\in\mathcal F$ and every $x\in\mathbb R^+$.